Concerning equations involving nonlocal operators, the regularity results for the fractional Laplacian problems are well understood. In particular, in [1]}, authors obtained the interior regularity results, while Ros-Oton and Serra in [2]}, proved the optimal boundary regularity. They proved that the weak solutions of the fractional Laplacian problem (i.e., \(p=q=2\)  and \(s_1=s_2\) ) with bounded right hand side and homogeneous Dirichlet boundary condition, are in \( C^{s}(\mathbb {R}^N)\) . One can rely on [3]} for regularity results of linear fractional problems involving a more general kernel. For the nonlinear and homogeneous operator case (\(p=q\ne 2\)  and \(s_1=s_2\) ), in [4]}, Di Castro et al. established the interior Hölder regularity results for nonlocal equations whose prototypes include the fractional \(p\) -Laplacian. Here, they established the Caccioppoli inequality and Logarithmic lemma and following De Giorgi's iteration technique, proved the local Hölder continuity result. In [5]}, Korvenpää et al. further obtained the interior and boundary Hölder regularity results for obstacle problems (also covering the Dirichlet problem) with \(zero\)  right hand side.
We also mention the work of M. Cozzi [6]}, for Hölder regularity results for minimizers of functional involving the fractional \(p\) -Laplacian energy in the fractional De Giorgi's class.
Using a slightly different approach, Iannizzotto et al. in [7]}, proved that the weak solutions of problem (REF ) with bounded right hand side, belong to the space \(C^{0,\alpha }(\overline{\Omega })\) , for some \(\alpha \in (0,s_1]\) . By constructing the barrier functions as a solution to some nonlocal equation on \(B_1\)  and scaling it appropriately to get a subsolution, they established weak Harnack type inequality to obtain the interior regularity results. Moreover, using the barrier arguments, in the spirit of Krylov's approach (as in [2]}), they proved the boundary behavior of the weak solutions.
Subsequently, in [9]}, Brasco et al. obtained the optimal Hölder interior regularity result and proved that the local weak solution \(u\in W^{s,p}_{\rm loc}(\Omega )\)  of problem (REF ) for the superquadratic and homogeneous case (i.e., \(2\le p=q\)  and \(s_1=s_2=s\) ) with \(f\in L^r_{\rm loc}(\Omega )\) , for suitable \(r>0\) , is in \(C^{\alpha }_{\rm loc}(\Omega )\) , for all \(\alpha <\min \lbrace 1,ps/(p-1)\rbrace \) . In this work, authors first obtain the improved regularity result for the case \(f=0\)  and then using a suitable scaling argument together with a decay transfer technique, they proved a similar result for non-zero \(f\) . Combining the boundary behavior of the solution from [7]} with the interior regularity result of [9]}, one can get the \(C^{s}\)  regularity result up to the boundary. We additionally refer to [12]}, for regularity results for nonlocal problems involving measure data. In [13]}, Iannizzotto et al. established a slightly different regularity result. Precisely, they extended the results of [2]} to the nonlinear setting and proved that the weak solution \(u\)  of problem (REF ), again for the case \(2\le p=q\) , \(s_1=s_2=s\in (0,1)\)  and bounded right hand side, satisfies \(\frac{u}{d^s} \in C^{\alpha }(\overline{\Omega })\) , for some \(\alpha \in (0,1)\) , where \(d(x):={\rm dist}(x,\partial \Omega )\) .
